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(3.4.) =0;>3VG=> H284:VABL v @CEC @V48=8 2 B>GFV  , (3.5) /:I> 225AB8 25:B>@ H284:>ABV DV;LB@0FVW V = (Vx,Vy,Vz) V 25:B>@ H284:>ABV @CEC @V48=8 v = (vx,vy,vz) , B>  (3.6) 0:>= 0@AV 0:>= 0@AV - >A=>2=89 70:>=, O:><C ?V4:>@OTBLAO @CE @V48=8 2 ?>@8AB><C A5@54>28IV. $@0=FC7L:8< V=65=5@>< 0@AV 2 1856 @. 5:A?5@8<5=B0;L=> 1C;> 2AB0=>2;5=>, I> H284:VABL DV;LB@0FVW ?@>?>@FV9=0 3@04VT=BC =0?>@C 9 A?@O<>20=0 C1V: 9>3> 7<5=H5==O. /:I> 225AB8 ?>=OBBO =0?>@C, I> C B5>@VW DV;LB@0FVW 287=0G0TBLAO @V2=VABN  (3.7) 45 p - B8A:,  IV;L=VABL @V48=8, g - ?@8A:>@5==O A8;8 2038, z - 35><5B@8G=0 28A>B0 =04 45O:>N 3>@87>=B0;L=>N ?;>I8=>N (2VAAN) ?>@V2=O==O,  = q - ?8B><0 2030, p/(g) - ? T7><5B@8G=89 =0?V@ (@V25=L 2>48 2 ? T7><5B@V), B>1B> C 25@B8:0;L=V9 B@C1FV, CAB0=>2;5=>W =86=V< :V=F5< C :@0?FV, 45 28<V@OTBLAO =0?V@ (<0;. 3.1); B> 70:>= 0@AV <>6=0 70?8A0B8 C 283;O4V @8A. 3.1.  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W =0720 287=0G0TBLAO DV78G=8< 7<VAB>< FVTW DC=:FVW, B><C I> 48D5@5=FV0;L=5 @V2=O==O ;V=VW B>:C <0T 283;O4 (3.18), O:8E <>6=0 70?8A0B8 2 B0:89 A?>AV1: . (3.26) 030;L=89 V=B53@0; FL>3> @V2=O==O T DC=:FVO (x,y) = C (C = const), >B65, =0 ;V=VOE B>:C DC=:FVO (x, y) 715@V30T ?>ABV9=5 7=0G5==O. 'OACT<> DV78G=89 7<VAB DC=:FVW ?>B>:C, 0 A0<5, ?>:065<>, I> DC=:FVO (x,y) ?>2'O70=0 7 ?>=OBBO< DV;LB@0FV9=>W 28B@0B8. 5E09 KL - 4>2V;L=0 :@820 2 >1;0ABV DV;LB@0FVW G - T =0?@O<=>W F8;V=4@8G=>W ?>25@E=V >48=8G=>W 28A>B8 7 CB2>@NNG>W, ?5@?5=48:C;O@=>W ?;>I8=8 xOy. 8B@0B0 @V48=8 Q G5@57 B0:C ?>25@E=N 4>@V2=NT AC<V DV;LB@0FV9=8E 28B@0B G5@57 =5A:V=G5==> <0;V 5;5<5=B8 :@82V9 KL. 024O:8 =5@>7@82=>ABV @>73;O=CB>3> ?>B>:C @V48=8 5;5<5=B0@=0 28B@0B0 d G5@57 5;5<5=B :@82>W dl 4>@V2=NT 0;351@0WG=V9 AC<V 28B@0B G5@57 4V;O=:8 1-2 9 2-3 - 2V4@V7:8 ?@O<8E, ?0@0;5;L=8E >AO< :>>@48=0B: d = dQx + dQy = vdl. C45<> 22060B8, I> 7=0G5==O 28B@0B8 Q(x,y) 7@>AB0T ?@8 @CAV C74>26 :@82>W KL C =0?@O<:C 2V4 B>G:8 1 4> B>G:8 3 (?>78B82=89 =0?@O<>: :@82>W). ">4V <0T<> d = dQx + sQy = vy(-dx) + vxdy ! d = d. =B53@CNG8 >AB0==T @V2=O==O C74>26 :@82>W 2V4 B>G:8 K 4> B>G:8 L, 7=0945<> HC:0=C DV;LB@0FV9=C 28B@0BC  (3.27) B>1B> 71V;LH5==O DC=:FVW ?>B>:C " C74>26 4>2V;L=>W :@82>W KL, C7OB>W 2 >1;0ABV DV;LB@0FVW G, 4>@V2=NT DV;LB@0FV9=V9 28B@0BV G5@57 FN :@82C. /:I> 7040B8 DC=:FVN ?>B>:C O: DC=:FVN 2V4 4>268=8 4C38 l :@82V9 KL, B> 4;O 287=0G5==O 28B@0B8 >45@68<>  (3.28) "><C I> DC=:FVW (x, y) V (x, y) 704>2>;L=ONBL C<>20< >HV - V<0=0, B> :><?;5:A=0 DC=:FVO  (3.29) 1C45 0=0;VB8G=>N 2 >1;0ABV DV;LB@0FVW G 9 WW <>6=0 @>73;O40B8 O: DC=:FVN :><?;5:A=>W 7<V==>W =f(z) , 45 z = x + iy . $C=:FVO (3.29) C B5>@VW DV;LB@0FVW =07820TBLAO :><?;5:A=8< ?>B5=FV0;>< DV;LB@0FVW. "0:8< G8=><, G5@57 :><?;5:A=89 ?>B5=FV0; DV;LB@0FVW 2AB0=>2;NTBLAO 72'O7>: DV;LB@0FVW 7 B5>@VTN DC=:FV9 :><?;5:A=>W 7<V==>W. >73;O=5<> I5 >4=C :><?;5:A=C 25;8G8=C vx - iyy , I> C B5>@VW DV;LB@0FVW =07820TBLAO :><?;5:A=>N H284:VABN DV;LB@0FVW. 8D5@5=FVNNG8 @V2=O==O (3.29) ?> z V 28:>@8AB>2CNG8 A?V22V4=>H5==O (3.20), (3.23), 7=0945<> ?>EV4=C . (3.30) "><C I> ?>EV4=0 0=0;VB8G=>W DC=:FVW T B0:>6 0=0;VB8G=>N DC=:FVTN, B> :><?;5:A=0 H284:VABL DV;LB@0FVW w, >1C<>2;5=0 @V2=VABN , (3.31) T 0=0;VB8G=>N DC=:FVTN 2 >1;0ABV DV;LB@0FVW G . 5><5B@8G=5 ?>40==O ?@> ?;>A:89 AB0;89 DV;LB@0FV9=89 ?>BV: 40T B0: =0720=C 3V4@>48=0<VG=C AVB:C, B>1B> AVB:C, CB2>@5=C AV<59AB2>< ;V=V9 ?>B>:C (x,y)=n=const V AV<59AB2>< 5:2V?>B5=FV0;L=8E ;V=V9 (x,y)= m=const, O:V >4=>G0A=> T 9 ;V=VO<8 @V2=8E =0?>@V2 h = hm = const.  C<>2 >HV- V<0=0 A;V4CT @V2=VABL  (3.32) O:0 ?>:07CT, I> 5:2V?>B5=FV0;L=V ;V=VW 9 ;V=VW ?>B>:C 270T<=> >@B>3>=0;L=V. "0:8< G8=><, O:I> 4;O 4>A;V46C20=>3> @CEC ?V475<=8E 2>4 7=09B8 :><?;5:A=89 ?>B5=FV0; DV;LB@0FVW (3.29) 01> :><?;5:A=C H284:VABL DV;LB@0FVW (3.31), B5 <>6=0 ;53:> 287=0G8B8 25;8G8=8 (x,y), (x,y), vx(x,y), vy(x,y), >B65 9 2AV V=HV E0@0:B5@8AB8:8 DV;LB@0FV9=>3> ?>B>:C. 5B>4 :>=D>@<=8E 2V4>1@065=L C B5>@VW DV;LB@0FVW /:I> 35><5B@8G=0 D>@<0 >1;0ABV G A:;04=0, B> 2V4HC:0==O @VH5==O :@09>2>W 7040GV ?>2'O70=5 7 1V;LH8<8 B@C4=>I0<8. "><C ?@8 28@VH5==V BVTW G8 V=H>W :@09>2>W 7040GV =0<030NBLAO A?@>AB8B8 O: 48D5@5=FV0;L=5 @V2=O==O V7 3@0=8G=8<8 C<>20<8, B0: V 284 >1;0ABV, C O:V9 2V4HC:CTBLAO 28@VH5==O. 4=8< 7 =09?>H8@5=VH8E <5B>4V2 B0:>3> A?@>I5==O :@09>2>3> 70240==O T <5B>4 ?5@5B2>@5==O =570;56=8E 7<V==8E (70<V=0 7<V==8E), 7>:@5<0, <5B>4 :>=D>@<=>3> ?5@5B2>@5==O =570;56=8E 7<V==8E. 5E09 C 45O:V9 >1;0ABV G =5>1EV4=> 7=09B8 @VH5==O :@09>2>W 7040GV 4;O @V2=O==O 0?;0A0 . (3.33) !?@>1CT<> A?@>AB8B8 284 >1;0ABV G 70 4>?><>3>N 70<V=8 7<V==8E  (3.34) 01>  (3.35) @8 ?5@5E>4V 4> =>28< 7<V==8E  V  <V=OTBLAO =5 BV;L:8 >1;0ABL G, 0;5 9 A0<5 48D5@5=FV0;L=5 @V2=O==O 9 3@0=8G=V C<>28. G5284=>, =091V;LH89 V=B5@5A ?@54AB02;ONBL ?5@5B2>@5==O, I> =5 <V=ONBL 284 48D5@5=FV0;L=>3> @V2=O==O, B>1B> 2 =0H><C 28?04:C ?5@5B2>@5==O, I>4> O:8E A0<5 @V2=O==O 0?;0A0 70;8H0TBLAO V=20@V0=B=8<. >:065<>, I> 2 FL><C 28?04:C DC=:FVW (3.34), I> 74V9A=NNBL ?5@5B2>@5==O >1;0ABV G C 1V;LH ?@>ABC >1;0ABL D, =0;560BL, O: V DC=:FVO (x,y), 4> :;0AC 30@<>=V9=8E DC=:FV9, 1V;LH5 B>3>, 2>=8 1C4CBL A?>;CG5=8<8 30@<>=V9=8<8 DC=:FVO<8. =0945<>     V4AB02;ONG8 7=0945=V 28@078 2 @V2=O==O (3.33), >45@68<> =0ABC?=5 48D5@5=FV0;L=5 @V2=O==O  . (3.36) G5284=>, 4;O B>3>, I>1 48D5@5=FV0;L=5 @V2=O==O (3.36) 1C;> @V2=O==O< 0?;0A0, =5>1EV4=>, I>1 ?5@5B2>@5==O (3.34) 704>2>;L=O;> B0:8< 28<>30<:  (3.37)  (3.38) . (3.39) V2=O==O (3.37) ?>:07CNBL, I> DC=:FVW  V  T 30@<>=V9=8<8 DC=:FVO<8. >74V;82H8 @V2=O==O (3.38) =0 , <0T<>  (3.40) V4AB02;ONG8 28@07 (3.40) C @V2=O==O (3.39), >45@6CT<> , (3.41) 72V4:8 <0T<>, I> O:I>  (3.42) B> = 1 9, >B65, 7 @V2=O=L (3.40) >45@68<> 01>  (3.43) 01>  (3.44) &V @V2=O==O T C<>20<8 >HV- V<0=0 9 ?>:07CNBL, I> DC=:FVW  V  T 30@<>=V9=8<8 DC=:FVO<8. 5@5B2>@5==O, 74V9A=N20=5 B0:8<8 DC=:FVO<8, ?5@52>48BL =5A:V=G5==> <0;V DV3C@8 ?;>I8=8 EC 2 ?>4V1=V W< DV3C@8 ?;>I8=8  70 C<>28, I> 28:>=CTBLAO (3.42). !0<5 B0:V ?5@5B2>@5==O 9 =07820NBLAO :>=D>@<=8<8. B65, O:I> ?5@5B2>@5==O, 74V9A=N20=5 DC=:FVO<8 (3.34), T :>=D>@<=8<, B> @V2=O==O (3.33) ?@89<5 284  01>   >AB0==L>W @V2=>ABV >45@6CT<>  (3.45) B@8<0=5 @V2=O==O B0:>6 T @V2=O==O< 0?;0A0, 45 G0AB:8 ?>EV4=V 28@060NBLAO G5@57 =>2V =570;56=V 7<V==V  V  - :>>@48=0B8 >1;0ABV D . "5?5@, O:I> CB2>@8B8 :><?;5:A=C DC=:FVN, C O:>W 4V9A=>N 9 CO2=>N G0AB8=0<8 T 2V4?>2V4=> DC=:FVW (x,y) V (x,y), B> B0:0 :><?;5:A=0 DC=:FVO =+i 1C45 0=0;VB8G=>N DC=:FVTN :><?;5:A=>W 7<V==>W z = x+iy, B>1B> (z)=(x,y)+i(x,y) = f(z). (3.46) /: <8 265 2V47=0G0;8, ?5@5B2>@5==O, 74V9A=N20=5 0=0;VB8G=>N DC=:FVTN (3.47), 01>, I> B5 6 A0<5, DC=:FVO<8 (3.34), =07820TBLAO :>=D>@<=8< CAN48 2 >1;0ABV G , 45 ?>EV4=0 =5 4>@V2=NT =C;N, B>1B> 45 28:>=CTBLAO C<>20  (3.47) "0:8< G8=><, @V2=O==O 0?;0A0 T V=20@V0=B=8< I>4> ?5@5B2>@5=L, 74V9A=N20=8E 0=0;VB8G=8<8 DC=:FVO<8 :><?;5:A=>3> 7<V==>3>. /:I> 6 ?5@5B2>@5==O (3.34) 74V9A=NTBLAO 4>2V;L=8<8 DC=:FVO<8 (x,y) V (x,y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x 9 y - :>>@48=0B B>G>: 28EV4=>W DV78G=>W >1;0ABV G . @8 @VH5==V :>=:@5B=8E DV78G=8E 7040G DC=:FVW  9  <0NBL ?52=C DV78G=C V=B5@?@5B0FVN. $V78G=0 ?>AB0=>2:0 7040G 287=0G0T 9 :@09>2V C<>28 4;O HC:0=8E DC=:FV9. 5B>4 :>=D>@<=8E 2V4>1@065=L 4>72>;OT B0:>6 C @O4V 28?04:V2, 0 A0<5, :>;8 3@0=8G=V C<>28 O: 4;O DC=:FVW (x,y), B0: V 4;O A?>;CG5=>W 7 =5N DC=:FVW (x,y), <0NBL A?5FV0;L=89 DV78G=89 7<VAB, 2V4HC:C20B8 @VH5==O @V2=O==O 0?;0A0 157?>A5@54=L>. # F8E 28?04:0E 4>A8BL 7=09B8 0=0;VB8G=C DC=:FVN, :>=D>@<=> DV78G=C >1;0ABL, I> 2V4>1@060T, G =0 >1;0ABL D 7<V=8 DV78G=8E  ?0@0<5B@V2 (x,y) V ?(x,y) . 84 >1;0ABV D 287=0G0TBLAO 3@0=8G=8<8 7=0G5==O<8 DC=:FV9 (x,y) V (x,y). ;O 70240=L ?;>A:>W DV;LB@0FVW, O:I> 240TBLAO :>=D>@<=> 2V4>1@078B8 >1;0ABL DV;LB@0FVW z =0 >1;0ABL :><?;5:A=>3> ?>B5=FV0;C  70 4>?><>3>N 45O:>W 0=0;VB8G=>W DC=:FVW  = f(z), B5 @>74V;82H8 4V9A=C 9 CO2=C G0AB8=8 DC=:FVW, I> 2V4>1@060T, 7=0945<> :><?;5:A=89 ?>B5=FV0;8 DV;LB@0FVW C 283;O4V  (3.48) 45 (x, y) - ?>B5=FV0; H284:>ABV DV;LB@0FVW, 0 (x, y) - DC=:FVO AB@C<C. @V< >?8A0=>W 0=0;VB8G=>W DC=:FVW - :><?;5:A=>3> ?>B5=FV0;C DV;LB@0FVW, C B5>@VW ?@>DV;L=>W DV;LB@0FVW @>73;O40NBLAO I5 42V 0=0;VB8G=V DC=:FVW: DC=:FVO C:>2AL:>3> G , I> 287=0G0TBLAO @V2=VABN  (3.49) V DC=:FVO C<5@>20, >1C<>2;5=0 @V2=VABN  (3.50) 45  - :V;L:VABL 2>48, I> =04E>48BL C @C=B ( > 0) 01> ?0@:>W ( < 0) 7 >48=8FV ?;>IV 3>@87>=B0;L=>W ?@>5:FVW 2V;L=>W ?>25@E=V 70 >48=8FN G0AC. "0:8< G8=><, :@09>25 70240==O B5>@VW ?;>A:>W AB0;>W 01> DV;LB@0FVW, I> :2078CAB0=>-28;8AL, ?>;O30T 2 BV<, I>1 4;O 7040=>W >1;0ABV DV;LB@0FVW Z 7=09B8 >4=C (01> 42V) 7 0=0;VB8G=8E DC=:FV9 (3.48),(3.49),(3.50). !?>AV1 02;>2AL:>3> !?>AV1 :>=D>@<=>3> 2V4>1@065==O 02;>2AL:>3> 70AB>A>2CTBLAO C 28?04:C, :>;8 2V4><0 3@0=8FO 28EV4=>W >1;0ABV DV;LB@0FVW G, I> 1C45<> ?>7=0G0B8 B0:>6 1C:2>N z (B><C I> >1;0ABL DV;LB@0FVW @>73;O40TBLAO 2 :><?;5:A=V9 ?;>I8=V z=x+ iy). V 2V4><0 >1;0ABL :><?;5:A=>3> ?>B5=FV0;C ()  (O:0 1C4CTBLAO 2 :><?;5:A=V9 ?;>I8=V = +i). ">4V E0@0:B5@8AB8G=0 DC=:FVO ?>B>:C z = F() 01> 72>@>B=0 W9 DC=:FVO - :><?;5:A=V ?>B5=FV0;8 H284:>ABV DV;LB@0FVW  = f(z) - 287=0G0TBLAO 2 @57C;LB0BV :>=D>@<=>3> 2V4>1@065==O >1;0ABV  =0 >1;0ABL z . 1;0ABL :><?;5:A=>3> ?>B5=FV0;C , O: ?@028;>, <>6=0 ?>1C4C20B8 BV;L:8 2 B><C 28?04:C, :>;8 3@0=8FO >1;0ABV DV;LB@0FVW z A:;040TBLAO 7 2>4>=5?@>=8:=8E V 2>4>?@>=8:=8E 4V;O=>:, B>1B> 3@0=8FO >1;0ABV DV;LB@0FVW A:;040TBLAO 7 5:2V?>B5=F80;L=8E ;V=V9 V ;V=V9 AB@C<C. # FL><C 28?04:C ?@><V6:8 28A>GC20==O 9 :@828E 45?@5AV9 2V4ACB=V (=0?V@=0 DV;LB@0FVO). "><C I> =0 5:2V?>B5=F80;L=8E ;V=VOE  = const, 0 =0 ;V=VOE AB@C<C  = const, B> >1;0ABL :><?;5:A=>3> ?>B5=FV0;C  C @>73;O=CB><C 28?04:C 702648 1C45 <0B8 283;O4 ?@O<>:CB=8:0 01> ?@O<>;V=V9=>3> 1030B>:CB=8:0, AB>@>=8 O:>3> ?0@0;5;L=V >AO< :>>@48=0B. 28G09=> 1C4CNBL 42V DC=:FVW, I> 2V4>1@060NBL: :>=D>@<=>, I> 2V4>1@060T =0 >1;0ABL DV;LB@0FVW z =86=N (01> 25@E=N) B0: =07820=C 4>?><V6=C =0?V2?;>I8=C  =  + i V :>=D>@<=> 2V4>1@060T =0   FN 65 4>?><V6=C =0?V2?;>I8=C . # FL><C 28?04:C @VH5==O 70240==O DV;LB@0FVW, B>1B> :><?;5:A=89 ?>B5=FV0; H284:>ABV DV;LB@0FVW (01> E0@0:B5@8AB8G=C DC=:FVN ?>B>:C), <>6=0 70?8A0B8 2 ?0@0<5B@8G=><C 284V z = f1(),  = f2(). (3.51) "><C I>  - ?@O<>;V=V9=89 ?@O<>:CB=8:, B> DC=:FVO  = f() 7=0E>48BLAO 70 4>?><>3>N V=B53@0;0 @VAB>D5;O-(20@F0. 3.4.2. !?>AV1 545@=8:>20-02;>2A:>3> # 28?04:C, :>;8 3@0=8FO >1;0ABV DV;LB@0FVW z <VAB8BL :@82V 45?@5AVW (B0: =0720=0 157=0?V@=0 01> 2V;L=0 DV;LB@0FVO), ?>;>65==O O:8E 70740;53V4L =52V4><>, :>=D>@<=5 2V4>1@065==O >1;0ABV DV;LB@0FVW z =0 >1;0ABL  01> =0?V2?;>I8=C  =5<>6;82>, E>G0 9 C FL><C 28?04:C, O: 9 C ?>?5@54=L><C, >1;0ABL  FV;:>< 287=0G5=0 9 <0T 283;O4 ?@O<>:CB=8:0 01> ?@O<>;V=V9=>3> 1030B>:CB=8:0. # 72'O7:C 7 @57C;LB0B0<8 . . 545@=8:>20 9 . . 02;>2AL:>3>, >B@8<0=8<8 =570;56=> 4@C3 2V4 4@C30, 1C2 70?@>?>=>20=89 A?>AV1, I> CAC20T B@C4=>IV, ?>2'O70=V 7 =5287=0G5=VABN ?>;>65==O :@82>W 45?@5AVW. !:>@8AB02H8AL 2V4><8<8 4;O DC=:FV9 (x, y) V (x,y) 3@0=8G=8<8 C<>20<8 =0 :@82>W 45?@5AVW BjCj  (3.52) 2>=8 70<VABL >1;0ABV 7<V==> z (>1;0ABL DV;LB@0FVW) 70?@>?>=C20;8 @>73;O40B8 >1;0ABL B0: =07820=>W DC=:FVW C:>2AL:>3> G, I> 287=0G0TBLAO @V2=VABN  (3.53) 01>  (3.54) "5?5@ <>6=0 70?8A0B8 3@0=8G=V C<>28 4;O DC=:FVW C:>2AL:>3>, 2V@=VH5, 4;O WW CO2=>W G0AB8=8: C74>26 3@0=8FV  7 25@E=L>N 2>4>9<>N (1'TD><)  (3.55)  AB =0 :@82>W 45?@5AVW  , @>7B0H>20=>W <V6 k-< 9 (k+1)-< 2>4>9<0<8,  (3.56) =0 3@0=8FV 7 (k + 1) -< 2>4>9<>N  (3.57) 45  - =02545=0 DV;LB@0FV9=0 28B@0B0 2 (k+1)-9 2>4>9<V; Q - ?>2=0 DV;LB@0FV9=0 28B@0B0. 0 2>4>=5?@>=8:=V9 4V;O=FV A1E1, =07820=V9 2>4>C?>@><, 7=0G5==O DC=:FV9 u 9 v =52V4><V, >4=0: <>6=0 2:070B8 <56V WE=L>W 7<V=8:  (3.58)  (3.59)  =5@V2=>ABV (3.59) 10G8<>, I> ;V=VO A1E1, I> T >1@07>< 2>4>C?>@0 2 ?;>I8=V DC=:FVW C:>2AL:>3> G, C:;045=0 2 3>@87>=B0;L=V9 A<C7V B>2I8=8 H. 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(3.60) /:I> 6 707=0G5=5 2V4>1@065==O 74V9A=NTBLAO G5@57 4>?><V6=C =0?V2?;>I8=C, B> HC:0=5 @VH5==O <>6=0 70?8A0B8 2 ?0@0<5B@8G=V9 D>@<V  (3.61) 8:;045=89 A?>AV1 <>6=0 70AB>A>2C20B8 9 C B><C 28?04:C, :>;8 2>4>?@>=8:=V 9 2>4>=5?@>=8:=V 4V;O=:8 =5 T 2V4?>2V4=> 3>@87>=B0;L=8<8 9 25@B8:0;L=8<8. @8 FL><C O: 28EV4=C >1;0ABL 4>A8BL 281@0B8 >1;0ABL DC=:FVW C:>2AL:>3> G, 0 ?VA;O   =0 >1;0ABL G 70 4>?><>3>N A?V22V4=>H5==O (3.53) 7=09B8 3@0=8FV >1;0ABV DV;LB@0FVW z (=0?V272>@>B=89 A?>AV1 545@=8:>20-02;>2AL:>3>). 1;0ABL G C FL><C 28?04:C 2818@0NBL B0:, I>1, 7 >4=>3> 1>:C, WW <>6=0 1C;> ?>@V2=O=> ;53:> :>=D>@<=> 2V4>1@078B8 =0 >1;0ABL  01> , 7 V=H>3> 1>:C - ?>1C4>20=0 4;O =5W >1;0ABL DV;LB@0FVW z ?>28==0 2V4?>2V40B8 @50;L=8< C<>20< DV;LB@0FV9=>3> 70240==O. >=D>@<=V ?5@5B2>@5==O 9 <>45;N20==O <0A> ?5@5=>AC @>F5A <V3@0FVW @>7G8==8E @5G>28= ?@8 DV;LB@0FVW ?V475<=8E 2>4, O: 2V4><>, >?8ACTBLAO A8AB5<>N @V2=O=L:  (3.62) ; (3.63)  (3.64) 01> 2 A:0;O@=V9 D>@<V  (3.65)  (3.66)  (3.67) 45 v = {vx,vy,vz} - 25:B>@ H284:>ABV DV;LB@0FVW, </4>1; (x,y,z,t) - ?>B5=FV0; DV;LB@0FVW; c(x, y, z, t) V N(x, y, z, t) - :>=F5=B@0FVO 48DC=4CNG>W @5G>28=8 2V4?>2V4=> 2 @V48==V9 V B25@4V9 D070E, 3/A< , D - :>5DVFVT=B :>=25:B82=>W 48DC7VW, <3/4>1;  - 0:B82=0 (01> 5D5:B82=0) ?>@8ABVABL A5@54>28I0, C O:><C ?@>BV:0T DV;LB@0FVO @>7G8=C; t - 3>48=0 (C 4>10E);  - >?5@0B>@ 0<V;LB>=0;  - AB0;0 H284:>ABV <0A>>1<V=C;  - :>5DVFVT=B @>7?>4V;C @5G>28=8 <V6; D070<8 2 C<>20E @V2=>2038 ?> ;V=V9=V9 V7B5@<V 5=@V cp = N. # 1030BL>E ?@0:B8G=8E 70240==OE <>6=0 >1<568B8AO 4>A;V465==O< ?@>F5AC <0A>?5@5=>AC @>7G8==8E C DV;LB@0FV9=><C ?>B>FV @5G>28= BV;L:8 =0 >A=>2V @V2=O=L, I> >?8ACNBL :>=25:B82=89 ?@>F5A, 0 A0<5:  (3.68)  (3.69) ?@8G><C <0A>>1<V= 287=0G0TBLAO B0:>N 4>A8BL @>7?>2AN465=>N 70;56=VABN:  (3.70) 45  - :>=F5=B@0FVO 3@0=8G=>W =0A8G5=>ABV. 02545=V @V2=O==O >?8ACNBL, O: ?@028;>, <V3@0FVN 9 DV78G=C B@0=AD>@<0FVN (A>@1FVN, 45A>@1FVN) :>=A5@20B82=8E 2>4>@>7G8==8E @5G>28=. /:I> 4>A;V48B8 <0A>?5@5=>A ?@8 ?;>A:>-25@B8:0;L=V9 V ?;0=>2V9 CAB0;5=V9 01> :207VCAB0;5=V9 DV;LB@0FVW ?V475<=8E 2>4, B> 4;O <>45;N20==O FL>3> ?@>F5A0 4>FV;L=> 70AB>AC20B8 :>=D>@<=5 ?5@5B2>@5==O @V2=O=L <0A>?5@5=>A0 4> :@82>;V=V9=8E 7<V==8E - :>>@48=0B0< B>G>: >1;0ABV :><?;5:A=>3> ?>B5=FV0;0 DV;LB@0FVW. # @07V ?;>A:>-25@B8:0;L=>W (?@>DV;L=>W) DV;LB@0FVW @V2=O==O @CE>2V ?V475<=8E 2>4 70?8HCBLAO C 283;O4V  (3.71) 45  - :>5DVFVT=B DV;LB@0FVW, h - =0?V@, O:89 287=0G0TBLAO @V2=VABN  (3.72) ?@8G><C 2VAL Oy A?@O<>20=> 25@B8:0;L=> 2=87, p - B8A:,  - IV;L=VABL, g - ?@8A:>@5==O A8;8 BO6V==O. # @07V ?;0=>2>W =0?>@=>W DV;LB@0FVW 2V4?>2V4=V @V2=O==O 70?8ACNBLAO B0::  = -Th;  (3.73) 0 2 @07V ?;0=>2>W 157=0?>@=>W DV;LB@0FVW   (3.74) # @V2=O==OE (3.73), (3.74) G5@57 T ?>7=0G5=> ?>BC6=VABL =0?V@=>3> 2>4>=>A=>3> H0@C: q - 25:B>@ ?8B><>W DV;LB@0FV9=>W 28B@0B8 (<2/4>1), a h - =0?V@, O:89 C 40=><C 28?04:C 287=0G0TBLAO B0:8< @V2=O==O<:  (3.75) 45 Z - 25@B8:0;L=0 :>>@48=0B0 B>G:8 DV;LB@0FV9=>3> ?>B>:C. @8?CI5==O, I> 4;O DV;LB@0FV9=8E B5GV9, I> @>73;O40NBLAO, <>6=0 ?>1C4C20B8 >1;0ABL :><?;5:A=>3> ?>B5=FV0;0  =  + i , 45  - DC=:FVO B5GVW, V I> 2V4><0 E0@0:B5@8AB8G=0 DC=:FVO B5GVW z = x + iy = F() = F1( , ) + i2( ,), (3.76) 70 4>?><>3>N 70<V=8 x = F1( ,); y = F2( ,) (3.77) ?5@5B2>@8<> @V2=O==O :>=25:B82=>W 48DC7VW 4> =>28E =570;56=8E 7<V==8E  V . # @57C;LB0BV B0:>W 70<V=8 @V2=O==O :>=25:B82=>W 48DC7VW 2 @07V ?;>A:>-25@B8:0;L=>W DV;LB@0FVW 70?8H5BLAO C 283;O4V  (3.78) C @07V ?;0=>2>W =0?>@=>W DV;LB@0FVW - C B0:><C 283;O4V:  (3.79) 0 2 @07V ?;0=>2>W 157=0?>@=>W DV;LB@0FVW ?5@5B2>@NTBLAO 4> 283;O4C  (3.80) /:I> 2 @V2=O==OE (3.78)-(3.80) D = 0, >B@8<0T<> @V2=O==O :>=25:B82=>3> <0A>?5@5=>AC 157 2@0E>2C20==O 48DC7V9=8E ?@>F5AV2, I> ?5@5B2>@5=V 4> =>28E 7<V==8E ,  01> ,  01> *, * 2V4?>2V4=> 4;O 28?04:V2 ?;>A:>-25@B8:0;L=>W, ?;0=>2>W =0?>@=>W V ?;0=>2>W 157=0?>@=>W DV;LB@0FVW, 0 A0<5:  (3.81)  (3.82)  (3.83) B65, C @07V =5EBC20==O 48DC7V9=8<8 ?@>F5A0<8 ?8B0==O ?@> 287=0G5==O :>=F5=B@0FVW @5G>28=, I> 701@C4=NNBL ?V475<=V 2>48, 72>48BLAO 4> @>72'O70==O 2V4?>2V4=>W DV;LB@0FV9=>W 7040GV B0 >4=>3> 7 @V2=O=L (3.81)-(3.83) V7 >4=>N 4>40B:>2>N (?>G0B:>2>N) C<>2>N, O:0 7040TBLAO 70;56=> 2V4 DV78G=>W ?>AB0=>2:8 7040GV. 06;82>N E0@0:B5@8AB8:>N ?@8 4>A;V465==V ?@>F5AC 701@C4=5==O ?V475<=8E 2>4 T G0A, ?@>BO3>< O:>3> 2 40=V9 B>GFV >1;0ABV  (01> >1;0ABV z) :>=F5=B@0FVO @>7G8==>W @5G>28=8 4>AO30T 287=0G5=>W 25;8G8=8. @V< B>3>, 28=8:0T ?8B0==O ?@> 287=0G5==O G0AC, ?@>BO3>< O:>3> :>=F5=B@0FVO @>7G8==>W @5G>28=8 4>AO30T 2 40=V9 B>GFV <0:A8<0;L=>3> 7=0G5==O. 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(284:VABL @>7?>2AN465==O @>7G8==>W C DV;LB@0FV9=><C ?>B>FV @5G>28=8 U(x,y,t) C 40=><C @07V 4>@V2=NT 4V9A=V9 H284:>ABV @CEC ?V475<=8E 2>4 V(x,y,t), O:0 72'O70=0 7V H284:VABN DV;LB@0FVW v(x,y,t) A?V22V4=>H5==O<  (3.84) 45 G5@57  ?>7=0G5=0 0:B82=0 ?>@8ABVABL @C=BC (?>@>48). 0 <8BBT2> ?@>BV:0NG8E A>@1FV9=8E ?@>F5A0E, I> 287=0G0NBLAO @V2=VABN (3.70), 0:B82=0 ?>@8ABVABL 70<V=NTBLAO 5D5:B82=>N ?>@8ABVABN A5@54>28I0, I> 287=0G0TBLAO @V2=VABN  (3.85) 7 (3.84) >B@8<CT<>  (3.86) VA;O ?5@5B2>@5==O @V2=>ABV (3.86) 4> =>28E =570;56=8E 7<V==8E  B0  <0T<>  (3.87) 0<VABL @V2=O=L (3.81)-(3.83) 7@CG=> @>73;O40B8 @V2=O==O  (3.88) 45  - 157@>7<V@=V 25;8G8=8, ?@8G><C . > @V2=O==O (3.88) ;53:> 725AB8 :>6=5 7 @V2=O=L (3.81)-(3.83). 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